In this scenario, Senate Minority Leader Bob Dole is the Republican presidential nominee of 1992 but loses to the Democratic nominee. There are many ways to achieve this. The first one I could think of is a Bush presidency from 1985-1993. Depending on who Bush's vice president is (possibly Jack Kemp), depends on whether Dole can defeat them and win the nomination. The other way I thought of is having a Democrat (possibly Al Gore) win the United States Presidential Election of 1988 and defeat Dole in 1992. In such scenarios, how do you think the Republican Party Presidential primaries of 1996 would look without the 'favorite son'? Pat Buchanan would probably not be the nominee because he was seen as too far to the right. Do you think George W. Bush would give the nomination a shot despite being Governor for such a short time?
Really, the simplest way to have Bob Dole not be a GOP presidential candidate in 1996 is to have him die in a traffic accident in, say, 1995. Then you eliminate all the countless effects (including "butterflies") of having GHW Bush instead of Ronald Reagan as president in 1985-89, or having a Democrat as president in 1989-93.
As to who the GOP nominates in such an event, I would think Lamar Alexander has the best chance. In OTL, he finished a close third in both Iowa and New Hampshire,
https://en.wikipedia.org/wiki/Republican_Party_presidential_primaries,_1996 and my guess is that in this ATL he would get enough of Dole's OTL voters to win both these early events, which would make him the clear front-runner. He would go on to lose to Clinton in November--it's hard to beat an incumbent president when there's peace and prosperity.